# theory of relativity

## A tip on speed of light units

A major advantage of Physics in terms of speed-of-light unit is we do NOT have to deal with huge numbers if we are to make a down to the pen and paper calculation. This is possible because speed of light is a constant even though a large number. So you can compute everything as if speed-of-light = 1 and the numbers and the related parameters such as what are known as Lorentz factors can be dealt like small numbers, especially the rapidity factor beta.

There is other definitions for rapidity factor? I think its just in terms of hyper-trigonometric functions, thats all.

For gamma you can expand the gamma in terms of beta, beta always runs from 0 to 1. Then gamma will be a binomial expansion. The binomial expansion is credited to Newton, what a man for Physics and Mathematics at the same time.

To get gamma there are 2 cases; ( — one ) you can either expand to a small number of power of beta; if the beta is small or ( — two ) if it ( — beta ) is close to 1, which is when people say ” we are dealing with an ultra-relativistic particle here ” you can do a trick. Chose a good power of beta, large enough so that you do not incur ( — more than necessary ) round off errors in your binomial expansion, which is not a mathematical problem as much as it may sound like, nonetheless it is, but take help from your Physics intuition if you are to enjoy certain privilege, you do not have to carry everything like a computer does.

This is of-course dependent on the physics problem.

## The relation between 3 anomalies of OPERA, FLYBY and Pioneer.

The very nature of earth-based gravitational effects is, at-least a fallout in the order of mm/second and application of Quantum Mechanics in case of OPERA apparatus brings out this inherent truth clearly.

One would see at minimum a (+ or -) mm/second order excess beyond speed-of-light = 1, no matter what. On the other hand flyby “anomaly” (which is rather a puzzle) is consistent with basic Relativity without need of Quantum Mechanics. The OPERA “anomaly” (which is basically a puzzle created out of carelessness) is based on an observation or more correctly a “misinterpretation” of 7.5 km/second but using Quantum Mechanics one sees that one should only observe ~5 mm/second given we have a fixed uncertainty on time-measurement to 1 nano-second level.

## GPS time measurement is not affected by Earth’s Orbital motion around sun.

This is an original research work of Manmohan Dash, published here, 30 November 2011, 1.20 pm.

Why orbital motion of earth around sun, does not matter to GPS?

GPS-IIRM

Because, it is not only earth, the satellite is also marching in a circle around the Sun and the Sun is simply too far.

Sun and earth distance is 1 AU ~ 149,597,871 kms — 15 karod kms, mere pyare bhaion aur behenon.

Earth-Satellite distance is about 26,000 kms = 0.000174 AU ~2 milli AU.

The angular speed of earth and satellite, due to Sun, will be almost same; w = r^(-1.5) = sqrt (1/ r.r.r), what difference it makes, if we change r, by 0.0002, when r is 1, the result is 1 changes to 0.9997.

So, if the angular speed of earth, is 1, the angular speed of satellite, assuming it is 26000 kms closer to sun-center, would be 0.9997. Thats a difference in the 4th digit. Now angular speed of earth itself is of 10^-5 order. So, this is 10^-9 order change. Why do we worry when earth spin itself is 1% of static effect, which is 1000 times at-least, smaller, than 1 nanosecond.

In other words, Sun’s effect, will be slightly more or less, between earth and satellite, by 10^-14, a digit in the 14th figure. If we adjust our digits, it will come to order 13 or go to order 15. So, we can safely say, the effect of Sun because of earth and satellite orbiting about Sun is, < 10^-12 nano-second = 10^-21 seconds. Anything at 10^-8 seconds is important.

## The physical meaning of Schwarzschild Radius. Flyby, Pioneer and OPERA anomaly.

since we are using light-speed in our theory we must make an error at the order of S-radii since by definition S-radii is the distance over which the relevant-mass does not let even light escape (read the linked article above and here: All masses behave like black-holes to some extent). SO thats (S-radii) the minimum error or speed anomaly. That is where the singularity of Theory of Relativity exists. Sun’s S-radii is 3 kms hence 3 kms/sec order error on satellites is permitted by Theory of Relativity, at the surface or away from surface the anomaly gradually drops down, as we go far from the effect of gravity, asymptotically to zero. Which is what we are seeing, the speed-anomaly of pioneer is decreasing. Except we do not go close to Sun’s surface hence we do not see 3 kms/sec. We are far away from it and seeing 1.27 mm/sec. (That is our radar is not placed on Sun’s surface but earth’s surface)

400 kms/year is not small, this millimeter accuracy is inbuilt into theory of Relativity, or it would not explain cosmology.

## The different kinds of time dilation, invariance of proper-time and principle of equivalence

For 2nd effect if the speed is negative that is source and observer are in relative approach the effect will always be anti-red shifted or as is known violet-shifted, again a distribution in angle which is extended towards translational vantage and line of sight vantage. (perpendicular to line of sight that is) The 3rd effect